For budget stays in Europe i recomend using youth hostels. We are British ex-pats living in the USA but travel on vacation to UK every year. My wife and I and our two teenage daughters have stayed at many youth hostels in the UK and Europe.
Its considerably less expensive than hotels and most definitely offers a superior social experience. Many youth hostels in the UK are converted historic buildings each with their own story to tell. Our most memorable experiences have been at hostels in Englands Lake District. It is also a very good way to go in major cities such as London or Paris where accomodation can be extemely expensive. In London we stayed a stones throw away from St Pauls, smack in the city center, in what used to be in days gone by the Choirboys residence now converted to a Youth Hostel, we were there for 5 days in a family room for 4 at a cost of around $150 a night. Typically though it would be around $100 per night (for 4) for similar facilities in more rural areas outside of the cities.
At first our daughters were hesitant about using youth hostels, but it is their preference now as they get old enough to travel on their own. They are very affordable, very safe and most are convenient to get to using public transport particularly in major towns and cities.
Many of the hostels have family rooms or single or double rooms and for less money you can share with other hostellers, we havent done that but my daughters have , its much like the dorm room experience at college. Almost all hostels have self catering kitchens which is very handy and also saves on costs.
More info can be gotten from http://www.yha.org.uk/ or if you are in the USA http://www.hihostels.com/ Both of these sites are affiliated with each other so if u have membership in one it is honored by the other. If u book through either, all that is required is a membership and 10% down payment and you pay the rest when you get there.